## About

46

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Introduction

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September 1999 - present

## Publications

Publications (46)

We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multiva...

We derive the connection relations between three q-Taylor polynomial bases. We also derive the connection relations between two of these bases and the continuous q-Hermite polynomials, as well as several new generating functions for the continuous q-Hermite polynomials. Our results also lead to some new integral evaluations. Using one of these conn...

Two seemingly disparate mathematical entities - quantum Bernstein bases and hypergeometric series - are revealed to be intimately related. The partition of unity property for quantum Bernstein bases is shown to be equivalent to the Chu-Vandermonde formula for hypergeometric series, and the Marsden identity for quantum Bernstein bases is shown to be...

We introduce the ( q , h )-blossom of bivariate polynomials, and we define the bivariate ( q , h )-Bernstein polynomials and ( q , h )-Bézier surfaces on rectangular domains using the tensor product. Using the ( q , h )-blossom, we construct recursive evaluation algorithms for ( q , h )-Bézier surfaces and we derive a dual functional property, a Ma...

We construct a q-analog of the blossom for analytic functions, the analytic q-blossom. This q-analog also extends the notion of q-blossoming from polynomials to analytic functions. We then apply this analytic q-blossom to derive identities for analytic functions represented in terms of the q-Poisson basis, including q-versions of the Marsden identi...

We introduce a blossoming procedure for polynomials related to the Askey–Wilson operator. This new blossom is symmetric, multiaffine, and reduces to the complex representation of the polynomial on a certain diagonal. This Askey–Wilson blossom can be used to find the Askey–Wilson derivative of a polynomial of any order. We also introduce a correspon...

We investigate some combinatorial and analytic properties of the n-dimensional Hermite polynomials introduced by Hermite in the late 19-th century. We derive combinatorial interpretations and recurrence relations for these polynomials. We also establish a new linear generating function and a Kibble–Slepian formula for the n-dimensional Hermite poly...

We introduce the G-blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G=G(t). By invoking the G-blossom, we construct G-Bernstein bases and G-Bézier curves and study their algebraic and geometric properties. We show that the G-blossom provides the dual fu...

We derive two new versions of Cooper's formula for the iterated Askey–Wilson operator. Using the second version of Cooper's formula and the Leibniz rule for the iterated Askey–Wilson operator, we derive several formulas involving this operator. We also give new proofs of Rogers' summation formula for series, Watson's transformation, and we establis...

Generalized quantum splines are piecewise polynomials whose generalized quantum derivatives agree up to some order at the joins. Just like classical and quantum splines, generalized quantum splines admit a canonical basis with compact support: the generalized quantum B-splines. Here we study generalized quantum B-spline bases and generalized quantu...

The -Bernstein-Bézier curves are generalizations of both the h-Bernstein-Bézier curves and the q-Bernstein-Bézier curves. We investigate two essential features of -Bernstein bases and -Bézier curves: the variation diminishing property and the degree elevation algorithm. We show that the -Bernstein bases for a non-empty interval satisfy Descartes' l...

We derive explicit formulas for the generating functions of the q-Bernstein basis functions in terms of q-exponential functions. Using these explicit formulas, we derive a collection of functional equations for these generating functions which we apply to prove a variety of identities, some old and some new, for the q-Bernstein bases.

We establish integral representations of Heine type for certain integrals of determinants. We use these representations to derive monotonicity properties of such determinants. New integral representations of Heine type for biorthogonal functions obtained from the general Gram–Schmidt orthonormalization process are given. We also establish the monot...

We consider two types of Hermite polynomials of a complex variable. For each type we obtain combinatorial interpretations for the linearization coefficients of products of these polynomials. We use the combinatorial interpretations to give new proofs of several orthogonality relations satisfied by these polynomials with respect to positive exponent...

We derive the asymptotics of certain combinatorial numbers defined on multi-sets when the number of sets tends to infinity but the sizes of the sets remain fixed. This includes the asymptotics of generalized derangements, numbers related to k-partite graphs, and exponentially weighted derangements. The asymptotics use integral and sum representatio...

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of clas...

We give a general method of characterizing symmetric orthogonal polynomials through a certain type of connection relations. This method is applied to Al-Salam–Chihara, Askey–Wilson, and Meixner–Pollaczek polynomials. This characterization technique unifies and extends some previous characterization results of Lasser and Obermaier and Ismail and Obe...

We introduce a new variant of the blossom, the q-blossom, by altering the diagonal property of the standard blossom. This q-blossom is specifically adapted to developing identities and algorithms forq-Bernstein bases and q-Bézier curves over arbitrary intervals. By applying the q-blossom, we generate several new identities including an explicit for...

A new variant of the blossom, the h-blossom, is introduced by altering the diagonal property of the standard blossom. The significance of the h-blossom is that the h-blossom satisfies a dual functional property for h-Bézier curves over arbitrary intervals. Using the h-blossom, several new identities involving the h-Bernstein bases are developed inc...

We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion co...

Developing computational models paves the way to understanding, predicting, and influencing the long-term behavior of genomic regulatory systems. However, several major challenges have to be addressed before such models are successfully applied in practice. Their inherent high complexity requires strategies for complexity reduction. Reducing the co...

In this work we apply a q-ladder operator approach to orthogonal polynomials arising from a class of indeterminate moment problems. We derive general representation of first and second order q-difference operators and we study the solution basis of the corresponding second order q-difference equations and its properties. The results are applied to...

We find a sequence of points {xn} in the interval [-1, 1] for which the corresponding sequence of Lagrange interpolation polynomials to the function |x| converges to |x| uniformly on the interval [-1, 1]. We also estimate the rate of convergence.

A q-version of a Bernstein basis of bivariate polynomials and a system of q-orthogonal bivariate polynomials over triangular domains are introduced to construct q-versions of the de Casteljau and the degree elevation algorithms. The polynomials are similar to the polynomials introduced recently by Farouki, Goodman, and Sauer [2]. The orthogonal pol...

In this paper we give upper bounds for a certain terminating 4.Fa series. Our estimates confirm special cases of a conjecture of Kresch and Tamvakis. We also give asymptotic estimates when the parameters in the 4Fa series are large, and they confirm the same conjecture.

In this paper we give upper bounds for a certain terminating
${}_4F_3$ series. Our estimates confirm special cases of a
conjecture of Kresch and Tamvakis. We also give asymptotic
estimates when the parameters in the
${}_4F_3$ series are large, and they confirm the same
conjecture.

The main objective of this work is to develop a methodology for risk management in a distributed system. Security is a very important issue when different users have potential access to operations of various databases of a system. There are benefits and risks involved in allowing these accesses overtime. Assuming that the probability of a user bein...

The purpose of this chapter is to present the key properties of fuzzy logic and adaptive nets and demonstrate how to use these, separately and in combination, to design intelligent systems. The first section introduces the concept of fuzzy sets and their basic operations. The t and s norms are used to define a variety of possible intersections and...

We find the spectrum ofthe inverse operator ofthe q- difference operator Dq,xf (x )=( f (x) − f (qx))/(x(1 − q)) on a family of weighted L2 spaces. We show that the spectrum is discrete and the eigenvalues are the reciprocals ofthe ze- ros ofan entire f unction. We also derive an expansion ofthe eigenfunctions of the q-difference operator and its i...

We prove that any set of polynomials orthogonal with respect to a discrete measure supported on equidistant points contained in a half line satisfy a second order difference equation. We also give a discrete analogue of the discriminant and give a general formula for the discrete discriminant of a discrete orthogonal polynomial. As an application w...

We study the minimization problem for weighted logarithmic energy integrals over the set of probability Borel measures supported on a closed subset of the extended complex plane. The weight is a nonnegative upper semi-continuous function that behaves like 1/|z| at infinity. We show that there exists a unique measure that minimizes the energy integr...

We obtain the zero distribution of sequences of classical
orthogonal polynomials associated with Jacobi, Laguerre, and
Hermite weights. We show that the limit measure is the extremal
measure associated with the corresponding weight.

In this work we prove that the associated polynomials of general q-orthogonal polynomials satisfy a fourth order q-difference equation. We provide two algorithms for constructing this equation and we identify its solution basis.

We find the spectrum of the inverse operator of the q-difference operator D-q,D-x f (x) = (f (x) - f (qx))/(x (1-q)) on a family of weighted L-2 spaces. We show that the spectrum is discrete and the eigenvalues are the reciprocals of the zeros of an entire function. We also derive an expansion of the eigenfunctions of the q-difference operator and...

Approximation by weighted rationals of the form w(n)r(n), where r(n) = p(n)/q(n), p(n) and q(n) are polynomials of degree at most [alpha n] and [beta n] respectively, and w is an admissible weight, is investigated on compact subsets of the real line for a general class of weights and given alpha greater than or equal to 0, beta greater than or equa...

We determine the class of functions, the divided difference of
which, at n distinct numbers, is a continuous function of the
product of these numbers.

Let w : S ? [0, 8) be a weight function on a set S ? R. We assume that the associated extremal measure µ? has density function v?(t) with finitely many singularities of logarithmic type. We show that any continuous function f on S which vanishes outside the set where v? is positive or has a logarithmic singularity, is the uniform limit on S of a se...

Ponzano and Regge conjectured certain asymptotic formulae for Racah coefficients with large angular momenta and gave convincing numerical evidence of their validity. We provide rigorous proofs of their asymptotic formulae. We use generating functions to derive an integral representation of Racah polynomials and then apply complex analytic methods t...

We determine the strong asymptotics for the class of Krawtchouk polynomials on the real line. We show how our strong asymptotics describes the limiting distribution of the zeros of the Krawtchouk polynomials.

We consider two problems concerning uniform approximation by weighted rational functions {wnrn}∞n=1, wherern=pn/qnhas real coefficients, deg pn⩽[αn] and deg qn⩽[βn], for givenα>0 andβ⩾0. Forw(x) :=exwe show that on any interval [0, a] witha∈(0, â(α, β)), every real-valued functionf∈C([0, a]) is the uniform limit of some sequence {wnrn}. An implicit...

We define a family of positive linear integral operators with kernels based on the Jacobi polynomials. Direct and inverse
approximation results for these operators are derived. The operators introduced are invariant under differentiation.
KeywordsPositive linear integral operator-Jacobi polynomial-direct approximation-inverse approximation-modulus...

## Projects

Project (1)